What is the output connected to? Sound card, amplifier, powered speaker?
Whatever it is, if it had infinite resistance to ground then changing the pot can change neither the current flowing out of the oscillator nor the voltage at the oscillator output. So the oscillator would not be affected.
Since it is, that implies the downstream resistance to ground is small enough to affect the oscillator. When you change the pot you change the load resistance.
In your oscillator is the output resistor connected between the pitch pot and the transistor emitter, or between the LED and the transistor collector? Last time I built one of these the former was about 400 mV peak to peak on a 7 V offset while the latter was 100 mV on a 700 mV offset. In the former case you’re well above the diode forward voltage and the diode has only a small effect on current and voltage. In the latter case you’re near or below the forward voltage, very little current gets through the diode, and the output voltage is close to zero. Either way I wouldn’t expect a diode to fix the problem you mention.
The usual way to isolate a signal from the effects of a variable load resistance is to buffer it with an op amp.
Added: Oh, what’s the value of the output attenuator? The smaller it is the less the downstream device will affect it. However, the smaller it is, the smaller the output voltage will be since it makes a divider with the oscillator’s 100k.